Finite-difference solutions of the x-tau wave equation

In a general inhomogeneous medium, finite difference is the most practical method for solving the wave equation. Despite its enormous computational cost, finite-difference schemes provide a comprehensive solution of the wave equation, which includes an accurate representation of amplitude.

In this example, we use the second-order acoustic wave equation for VTI media in $(x-\tau)$-domain, given by equation (27) and therefore need to solve simultaneously

$$\begin{eqnarray} \frac{\partial^2 P}{\partial t^2} =-8\,\frac{\partial^4 F}{\par... ...v^2}\,\left( 1 + 2\,\eta \right) \,{{\sigma }^2} \right) +f,\,\,\,\end{eqnarray}$$ (28)

and

$$P = \frac{\partial^2 F}{\partial t^2},$$

where $f(x,\tau)$ is the forcing function. We use a second-order finite-difference approximation for P-derivatives in equation (28) and a fourth-order approximation for F-derivatives. The solution for elliptically anisotropic media is obtained by setting $\eta$=0. Since Alkhalifah (1997b) discusses in detail finite-difference application to a fourth-order equation closely resembling this one, no detailed discussion is included here.

 

velwave
Figure 7 Velocity models in the conventional depth domain (top), and in the $(x-\tau)$-domain (bottom). The velocity model includes a negative velocity anomaly perturbed from a background medium with v(x)=2000+0.4x m/s.

Figure 7 shows a velocity model in depth (on the top), and its equivalent mapping in time (bottom). Figure 8 shows the wavefield at 0.65 s resulting a source igniting at time 0 s, that corresponds to the isotropic velocity model in Figure 7. The wavefield is computed using finite-difference approximations of equation (26). The velocity model given in the $(x-\tau)$-domain is the input velocity model in the finite-difference application. This same velocity model is used to map the wavefield solution back to depth. The solid curves in Figure 8 show the solution of the conventional eikonal solver Vidale (1990) implemented in the depth domain, and these curves nicely envelope the wavefield solution. Therefore, computing the wavefield in the $(x-\tau)$-domain and in the conventional depth domain are equivalent, regardless of the lateral inhomogeneity. However, the $(x-\tau)$-domain implementation becomes independent of vertical P-wave velocity when $\frac{d \alpha}{d x}=0$.

 

wavepr
Figure 8 Top: The wavefield in the $(x-\tau)$-domain at 0.65 s resulting from a source at distance 2000 m and $\tau$=0 for the isotropic velocity model shown in Figure 7. Bottom: the same wavefield solution after mapping back to depth using the same velocity model. The black curve is the solution of the eikonal equation for the velocity model in Figure 7 implemented using the conventional depth-domain eikonal solver.

Is is also important to note that the apparent frequency of the time section is velocity independent, while waves in the depth section have wavelengths very much dependent on velocity.